
We note that the norm constraint here has dimension \(dc\)  and \(\mathbf {vec}(\mathbf {W}_1)\)  has dimension \(s^2\) , so by [1]}, [2]} this strong duality result from Lemma REF  requires that \(m^* \le n\min \lbrace s^2, dc\rbrace \) . Now, let \(\mathbf {D}_j^{(i)} \in \mathbb {R}^{sd \times sd}\)  be the \(i\) th block of \(\mathbf {D}_j\) , and enumerate over all possible hyperplane arrangements \(j\) . Then, we have
\(p_{MM}^* &= \max _{\mathbf {V}_i} -\sum _{i=1}^n \mathcal {L}^* \left(\mathbf {V}_i,\mathbf {Y}_i \right) \nonumber \\\text{s.t.} &~ \max _{\begin{array}{c}\Vert \mathbf {w}_1\Vert _2\le 1 \\ j \in [P] \\ \mathbf {K}_j\mathbf {w}_1 \ge 0\end{array}} \left\Vert \sum _{i=1}^n (\mathbf {I}_d \otimes \mathbf {V}_i^T) \mathbf {D}_j^{(i)} ({\mathbf {X}}_i^\top \otimes \mathbf {I}_{s}) \mathbf {w}_1\right\Vert _2\le \beta .\) 
